Stochastic fluctuations around two different dominant reaction pathways in a two-dimensional transition. As the temperature rises, the stochastic fluctuations become large and eventually the two paths become statistically indistinguishable.
The total time for a transition from x i = −1 to x f = 1 in the quartic potential (45), calculated using Eq. (24) with different number of discretization steps, in the original theory and in the EST.
Comparison between the DRP expression for the conditional probability for the diffusion in a harmonic potential, in a system of units in which α = 4, β = 10, and Dβ = 1. In the left panel we show the DRP and the corresponding exact conditional probability P(x f , t|x i ) as function of the final position x f , for x i = 0 and for two times t = 0.1 and t = 1. We also show the corresponding Boltzmann equilibrium distribution. In the right panel, we compare the DRP prediction for the probability density and for the probability current with the corresponding exact results, as a function of the time interval t, at a fixed final position x f = 0.2.
The definition of ∂W as the hyper-surface orthogonal to the dominant path the the point solution of the transition state Eq. (27).
Left panel: The absolute value of the normalized probability current at the sadde-point x f = 0, evaluated with the DRP method at different times t, for x i = −ω and β = 5. The different total times are determined by different choices of the effective energy parameter. Righ panel: The ratio between the reaction rates evaluated with the DRP method and with Kramers theory.
Left panel: The contour plot of the potential energy [Eq. (78)]. Right panel: the ratio of the rates evaluated in the DRP approach and using Kramers theory, DRP calculations, and direct MD simulations.
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